let Y be non empty set ; :: thesis: for a, u being Function of Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for PA being a_partition of Y holds (All (a,PA,G)) 'imp' u '<' Ex ((a 'imp' u),PA,G)
let a, u be Function of Y,BOOLEAN; :: thesis: for G being Subset of (PARTITIONS Y)
for PA being a_partition of Y holds (All (a,PA,G)) 'imp' u '<' Ex ((a 'imp' u),PA,G)
let G be Subset of (PARTITIONS Y); :: thesis: for PA being a_partition of Y holds (All (a,PA,G)) 'imp' u '<' Ex ((a 'imp' u),PA,G)
let PA be a_partition of Y; :: thesis: (All (a,PA,G)) 'imp' u '<' Ex ((a 'imp' u),PA,G)
let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not ((All (a,PA,G)) 'imp' u) . z = TRUE or (Ex ((a 'imp' u),PA,G)) . z = TRUE )
A1: z in EqClass (z,(CompF (PA,G))) by EQREL_1:def 6;
assume ((All (a,PA,G)) 'imp' u) . z = TRUE ; :: thesis: (Ex ((a 'imp' u),PA,G)) . z = TRUE
then A2: ('not' ((All (a,PA,G)) . z)) 'or' (u . z) = TRUE by BVFUNC_1:def 8;
A3: ( 'not' ((All (a,PA,G)) . z) = TRUE or 'not' ((All (a,PA,G)) . z) = FALSE ) by XBOOLEAN:def 3;

now :: thesis: ( ( 'not' ((All (a,PA,G)) . z) = TRUE & (Ex ((a 'imp' u),PA,G)) . z = TRUE ) or ( u . z = TRUE & (Ex ((a 'imp' u),PA,G)) . z = TRUE ) )

per cases ( 'not' ((All (a,PA,G)) . z) = TRUE or u . z = TRUE ) by A2, A3;

case 'not' ((All (a,PA,G)) . z) = TRUE ; :: thesis: (Ex ((a 'imp' u),PA,G)) . z = TRUE

then consider x1 being Element of Y such that
A4: x1 in EqClass (z,(CompF (PA,G))) and
A5: a . x1 <> TRUE by BVFUNC_1:def 16;

now :: thesis: ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & (a 'imp' u) . x = TRUE )

assume for x being Element of Y holds
( not x in EqClass (z,(CompF (PA,G))) or not (a 'imp' u) . x = TRUE ) ; :: thesis: contradiction
then (a 'imp' u) . x1 <> TRUE by A4;
then (a 'imp' u) . x1 = FALSE by XBOOLEAN:def 3;
then A6: ('not' (a . x1)) 'or' (u . x1) = FALSE by BVFUNC_1:def 8;
( 'not' (a . x1) = TRUE or 'not' (a . x1) = FALSE ) by XBOOLEAN:def 3;
hence contradiction by A5, A6; :: thesis: verum

end;

hence (Ex ((a 'imp' u),PA,G)) . z = TRUE by BVFUNC_1:def 17; :: thesis: verum

end;

case A7: u . z = TRUE ; :: thesis: (Ex ((a 'imp' u),PA,G)) . z = TRUE

now :: thesis: ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & (a 'imp' u) . x = TRUE )

assume for x being Element of Y holds
( not x in EqClass (z,(CompF (PA,G))) or not (a 'imp' u) . x = TRUE ) ; :: thesis: contradiction
then (a 'imp' u) . z <> TRUE by A1;
then (a 'imp' u) . z = FALSE by XBOOLEAN:def 3;
then ('not' (a . z)) 'or' (u . z) = FALSE by BVFUNC_1:def 8;
hence contradiction by A7; :: thesis: verum

end;

hence (Ex ((a 'imp' u),PA,G)) . z = TRUE by BVFUNC_1:def 17; :: thesis: verum

end;

end;

end;

hence (Ex ((a 'imp' u),PA,G)) . z = TRUE ; :: thesis: verum